#499257

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,21)(12,17)(13,18)(14,16)(15,23)(19,27)(20,26)(22,31)(24,38)(25,30)(28,33)(29,39)(32,37)(34,40)(35,36);(2,5)(6,8)(9,11)(10,15)(12,22)(13,23)(14,24)(16,32)(17,30)(18,21)(19,29)(20,31)(25,26)(27,36)(28,37)(33,38)(34,35)(39,40);(10,16)(11,22)(12,26)(14,27)(15,29)(17,34)(18,33)(19,20)(21,35)(23,25)(24,37)(28,40)(30,32)(31,38);(1,3,4,7)(2,6,5,8)(9,12,13,26)(10,17,18,20)(11,22,23,25)(14,27,28,40)(15,30,21,31)(16,19,33,34)(24,36,37,39)(29,38,35,32)

Polycyclic presentation:

2^9|d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319899, #72319904, #72319933, …
maximal normal subgroup of	#72319899, #72319904, #72319933, …
maximal quotient of	C₂×#499257

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406453);